Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #481 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1926 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,5,11,8,23)(3,12,6,21,15,9,16,25)(4,17,18,28,20,32,19,24)(10,22,26,14,27,30,29,31);(1,3,8,16,5,15,7,6)(2,9,23,21,11,12,13,25)(4,14,18,30,20,31,19,22)(10,24,26,17,27,28,29,32);(1,4,7,19,5,20,8,18)(2,10,23,26,11,27,13,29)(3,14,6,22,15,31,16,30)(9,24,21,17,12,28,25,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=dg,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13140, #494883, #494999, … |
| maximal normal subgroup of | C2×#13140, #494883, #494999, … |
| maximal quotient of | #140118, #140119, #140144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6128);